[plus]

What I'm saying is true -- if the initial state is chosen to be a superposition of states, then the time-dependent Schrodinger equation enables one to simulate the time evolution of the electron density.

> thus, the "change" in the simulation posted here would be uninteresting since it would just correspond to clicking a different eigenstate in the top right!

That is a drastic oversimplification of reality. Spontaneous emission of a photon is not instantaneous. It is still possible to simulate the time-evolution of this process.

[ellis-bell]

everything is a simplification. of course it's not spontaneous, but that is literally the word used by physicists to describe the phenomenon.

what does the wave equation of the electron in a hydrogen atom look like during "spontaneous" emission of a photon? i don't think anyone has any idea.

i'm talking about something entirely separate from a linear combination of two energy eigenstates. i'm not saying take

\psi = \sin{\theta} \psi_1 + \cos{\theta} \psi_2

where \psi_1 and \psi_2 are eigenstates of the hydrogen atom hamiltonian and simulate it. i'm saying there is a phenomenon that i'm pretty sure wouldn't be adequately modelled by a smooth function.

edit: add explanation of \psi_1 and \psi_2

[plus]

The hydrogenic orbitals form a complete and orthonormal basis in which you can expand any arbitrarily-chosen one-electron wavefunction. That is to say that any arbitrarily chosen initial state for the one-electron wavefunction (and thus any arbitrarily chosen electron density) can be expressed as a linear combination of the hydrogenic orbitals. This, in combination with the time-dependent Schrodinger equation, enables us to simulate the time evolution of the hydrogenic system starting from any arbitrarily chosen initial state.

> what does the wave equation of the electron in a hydrogen atom look like during "spontaneous" emission of a photon? i don't think anyone has any idea.

It sounds like you are not aware that this is an entire field of research within the theoretical chemistry community. Theoreticians have been studying spectroscopy for a century.